The versification of the Hebrew bible in the BHS, Westminster Leningrad Codex, and the JPS Tanach is different than the versification of the Hebrew bible in Christian publications, such as the RSV/NSRV. (Verses at the Leviticus 5-6 chapter interface are one example.)

The simplest (and inaccurate) answer is that the versification of the Hebrew Bible is according to Jewish canonical tradition, and modern translations often (although not entirely) follow a Christian canonical tradition, with the classic manifestation being the Vulgate. However, the traditions are not monolithic. For example, the LXX (Jewish tradition) has versification variants, even within its sub-types. And a number of modern language translations follow the Jewish tradition. The Hungarian Karoli (equivalent to the English KJV), for example. Most English translations follow the Christian/Vulgate tradition.

The origins of the various versification systems are probably legendary. For one explanation, see this webpage.

I have a file that traces these differences, but unfortunately at the moment it is inaccessible due to a hard disk failure yesterday. However, I did find a comprehensive list here. You should find some form of the data in a useful format.

One type of "discrepancy" that occurs is in the Psalms, where the Psalm titles in Hebrew texts (omitted in some translations) are usually counted as the first verse. When they are used in translations, often the first group of lines following is counted as verse 1 instead. And some Psalm chapter divisions are different, too (e.g., 9 and 10 are counted as one chapter in the Hebrew texts), which affects the numbering of subsequent chapters. The LXX and Vulgate tend to differ from the Hebrew chapter divisions starting with 9-10. This affects the chapter numbering in many translations today, as well as the verse numbering.

For a Hebrew-only site like tanach.us, I see no reason to add the alternate versification except as a crutch for those who have no knowledge of Hebrew.

And for b-hebrew, one should always use the Jewish versification when citing the Hebrew text, and only use the alternate versification if citing a ancient version or modern translation.

The whole point of versification is to help identify the part of the text you want to talk about. So whatever the text/edition uses, that is the one you should use to help others find your text citation.

Yes, it's very complex. The Leningrad Codex is different not just from Aleppo, but from classic rabbinic manuscripts and early printed editions. Most of these differences have been noted, but not all in one place.